Basic Equations: Differential Form - Mechanics of Structures: Variational and Computational Methods - page 38

s 2

Maximum shear
stress criterion

s ys

von Mises
criterion

-s ys

s 1

s ys

-s ys

FIGURE 1.14
The von Mises ellipse and the maximum shear stress hexagon.

Figure 1.14 shows the plot of Eq. (1.96) and the corresponding expression for the maxi-
mum shear stress theory. Note that both theories intersect the axes at the same points. They
also have in common the two points of intersection with the bisectors of the first and third
quadrants. It is clear from the figure that the von Mises criterion is less conservative than
the maximum shear stress criterion. On one hand the maximum shear stress criterion is lin-
ear, or more accurately a set of bilinear curves, and is easy to manipulate, on the other hand
the von Mises criterion involves a single expression in contrast to the three expressions of
Eq. (1.93).

1.8 Engineering Beam Theory

The technical or engineering theories for structural members are distinguished from the
theory of elasticity in that their governing equations are highly tractable. This is primarily
the result of imposing simplifying geometric assumptions on the theory, e.g., for the bending
of beams it is assumed that planar cross-sections remain planar throughout the bending
process. In this chapter, it is shown that the engineering beam theory equations can be
developed and expressed in a fashion similar to the elasticity equations just considered.

1.8.1 Kinematical Relationships

The kinematical relationships are the strain-displacement equations. For a beam, the bend-
ing strain is taken to be the curvature is the radius of curvature of the beam

axis through the centroids of the cross-section. From analytical geometry, the definition of

κ = 1 /ρ , where ρ